Uncertainty quantification of the lattice Boltzmann method focussing on studies of human-scale vascular blood flow

Uncertainty quantification is becoming a key tool to ensure that numerical models can be sufficiently trusted to be used in domains such as medical device design. Demonstration of how input parameters impact the quantities of interest generated by any numerical model is essential to understanding the limits of its reliability. With the lattice Boltzmann method now a widely used approach for computational fluid dynamics, building greater understanding of its numerical uncertainty characteristics will support its further use in science and industry. In this study we apply an in-depth uncertainty quantification study of the lattice Boltzmann method in a canonical bifurcating geometry that is representative of the vascular junctions present in arterial and venous domains. These campaigns examine how quantities of interest—pressure and velocity along the central axes of the bifurcation—are influenced by the algorithmic parameters of the lattice Boltzmann method and the parameters controlling the values imposed at inlet velocity and outlet pressure boundary conditions. We also conduct a similar campaign on a set of personalised vessels to further illustrate the application of these techniques. Our work provides insights into how input parameters and boundary conditions impact the velocity and pressure distributions calculated in a simulation and can guide the choices of such values when applied to vascular studies of patient specific geometries. We observe that, from an algorithmic perspective, the number of time steps and the size of the grid spacing are the most influential parameters. When considering the influence of boundary conditions, we note that the magnitude of the inlet velocity and the mean pressure applied within sinusoidal pressure outlets have the greatest impact on output quantities of interest. We also observe that, when comparing the magnitude of variation imposed in the input parameters with that observed in the output quantities, this variability is particularly magnified when the input velocity is altered. This study also demonstrates how open-source toolkits for validation, verification and uncertainty quantification can be applied to numerical models deployed on high-performance computers without the need for modifying the simulation code itself. Such an ability is key to the more widespread adoption of the analysis of uncertainty in numerical models by significantly reducing the complexity of their execution and analysis.


High pressure cases
Supplementary Figures 14 -16 present the flow results from the coarse domain under the first three cases of boundary condition variation discussed in the main manuscript with pressure conditions a factor of 100 higher.Supplementary Figures 18 -21 present these higher pressure cases in the fine domain for all four boundary condition variations.The coefficients of variation calculated for these cases are presented in Supplementary Table 1.
In Supplementary Figures 14 and 15, where only one of the pressure outlets is being varied, it can be seen that the mean value of this changing pressure is the dominating factor in the variation of both the pressure and velocity quantities of interest.The area of exception is the velocity in the main channel prior to the bifurcation.Here the inlet velocity is the only factor influencing the measured quantity.Beyond the bifurcation, the impact of the inlet velocity is negligible.
When only the pressure outlets are being changed (Supplementary Figure 16), a high degree of symmetry between the analysed parameters is recorded.This would appear to indicate that for the chosen geometric configuration, the asymmetry of the domain does not impose a significant impact.This is potentially related to the flow distance between the point of observation and the pressure outlets being the same in this case.Before the bifurcation, the small variation of velocity within the channel is dominated by the amplitude of the oscillating boundary conditions.Beyond the bifurcation the combined effects of the chosen boundary conditions describes approximately 75% of the changes in velocity.The remaining 25% is almost fully accounted for by the mean pressure values suggesting that it is the combination of these that is also influential.
In the combined boundary case (Supplementary Figure 17) also records a combination of the observed statistical results.Before the bifurcation, the inlet velocity is solely responsible for the variation in the domain and less than 10% of pressure variance.The mean pressure of the outlet boundaries, either singularly or in combination, are responsible for over 90% of the change in pressure distribution throughout the whole domain.Beyond the bifurcation they are also responsible for the variation in velocity and mostly in a higher order combination.The distribution of velocity increases significantly once it becomes driven by the pressure boundary conditions.

Forearm arteries
In Supplementary Figure 22, the normal distribution of the mean and maximum velocities observed in the personalised vessels can be observed.This data has been sampled based on a third-order polynomial chaos expansion of the observed simulation data.Supplementary Figure 14: Uncertainty analysis of the pressure and velocity within the coarse bifurcation geometries determined with a third-order polynomial chaos expansion (BC1, higher pressure).
(a) Coarse -Pressure distribution (b) Coarse -Velocity distribution (c) Coarse -Pressure Sobol indices (d) Coarse -Velocity Sobol indices (e) Fine -Pressure distribution (f) Fine -Velocity distribution (g) Fine -Pressure Sobol indices (h) Fine -Velocity Sobol indicesSupplementary Figure1: Uncertainty analysis of the pressure and velocity within the main channel coarse and fine bifurcation geometries determined with a third-order polynomial chaos expansion.Algorithmic parameters of the LBM have been varied here with constant boundary conditions.
Main -Pressure Sobol indices (d) Main -Velocity Sobol indices (e) Branch -Pressure distribution (f) Branch -Velocity distribution (g) Branch -Pressure Sobol indices (h) Branch -Velocity Sobol indicesSupplementary Figure3: Uncertainty and parameter sensitivity analysis of the pressure and velocity within the bifurcation geometry determined with a fourth-order polynomial chaos expansion.Algorithmic parameters of the LBM have been varied here with constant boundary conditions.
Main -Pressure Sobol indices (d) Main -Velocity Sobol indices (e) Branch -Pressure distribution (f) Branch -Velocity distribution (g) Branch -Pressure Sobol indices (h) Branch -Velocity Sobol indicesSupplementary Figure4: Uncertainty and parameter sensitivity analysis of the pressure and velocity within the bifurcation geometry determined with a fifth-order polynomial chaos expansion.Algorithmic parameters of the LBM have been varied here with constant boundary conditions.
Branch -Pressure distribution (f) Branch -Velocity distribution (g) Branch -Pressure Sobol indices (h) Branch -Velocity Sobol indicesSupplementary Figure5: Uncertainty and parameter sensitivity analysis of the pressure and velocity within the bifurcation geometry determined with a sixth-order polynomial chaos expansion.Algorithmic parameters of the LBM have been varied here with constant boundary conditions.
Main -Pressure Sobol indices (d) Main -Velocity Sobol indices (e) Branch -Pressure distribution (f) Branch -Velocity distribution (g) Branch -Pressure Sobol indices (h) Branch -Velocity Sobol indicesSupplementary Figure6: Uncertainty and parameter sensitivity analysis of the pressure and velocity within the bifurcation geometry determined with a seventh-order polynomial chaos expansion.Algorithmic parameters of the LBM have been varied here with constant boundary conditions.
Main -Pressure Sobol indices (d) Main -Velocity Sobol indices (e) Branch -Pressure distribution (f) Branch -Velocity distribution (g) Branch -Pressure Sobol indices (h) Branch -Velocity Sobol indicesSupplementary Figure7: Uncertainty and parameter sensitivity analysis of the pressure and velocity within the bifurcation geometry determined with an eighth-order polynomial chaos expansion.Algorithmic parameters of the LBM have been varied here with constant boundary conditions.
Coarse -Pressure Sobol indices (d) Coarse -Velocity Sobol indices (e) Fine -Pressure distribution (f) Fine -Velocity distribution (g) Fine -Pressure Sobol indices (h) Fine -Velocity Sobol indicesSupplementary Figure8: Uncertainty analysis of the pressure and velocity within the main channel coarse and fine bifurcation geometries determined with a third-order polynomial chaos expansion.Algorithmic parameters of the LBM have been varied here with constant boundary conditions.
Main -Pressure Sobol indices (d) Main -Velocity Sobol indices (e) Branch -Pressure distribution (f) Branch -Velocity distribution (g) Branch -Pressure Sobol indices (h) Branch -Velocity Sobol indices Supplementary Figure 10: Uncertainty analysis of the pressure and velocity within the coarse bifurcation geometries determined with a third-order polynomial chaos expansion.Boundary conditions consistent with Figure 4 of the main manuscript.
Main -Pressure Sobol indices (d) Main -Velocity Sobol indices (e) Branch -Pressure distribution (f) Branch -Velocity distribution (g) Branch -Pressure Sobol indices (h) Branch -Velocity Sobol indicesSupplementary Figure12: Uncertainty analysis of the pressure and velocity within the coarse bifurcation geometries determined with a third-order polynomial chaos expansion.Boundary conditions consistent with Figure6of the main manuscript.
Main -Pressure Sobol indices (d) Main -Velocity Sobol indices (e) Branch -Pressure distribution (f) Branch -Velocity distribution (g) Branch -Pressure Sobol indices (h) Branch -Velocity Sobol indicesSupplementary Figure13: Uncertainty analysis of the pressure and velocity within the coarse bifurcation geometries determined with a third-order polynomial chaos expansion.Boundary conditions consistent with Figure7of the main manuscript.

Table 1 :
Coefficients of variation of input parameters and quantities of interest as generated by the LBM implemented within HemeLB for the high pressure boundary condition campaigns.